quently vis viva will constantly be generated, till, as the waves become extricated the one from the other, the vis viva will be, what it will ever after remain, exactly what it was at first. Hence, if it were true, as Dr. Helmholtz assumes, that the normal actions disappear from the equation of vis viva when we consider the internal motions of continuous masses, that fact alone would suffice to demonstrate the entire failure of his principle of the conservation of force in cases of motion such as that we have been considering. II. That the sum of the tensions does not in general vanish when a wave traverses a medium may be seen as follows: Let x and y be the ordinates (measured parallel to the axis of the tube) in the state of rest and at the time t of one surface of an element made by planes perpendicular to the axis, p the pressure at the same surface, D the density of equilibrium. The equation of motion may be put under the form whence, multiplying by dy dt and integrating, we get for the equa dt the integration being effected between any limits we may fix upon. Suppose that we have two waves of condensation, such as those already described, travelling in opposite directions; and (a and y being measured parallel to c AC in the annexed figure) let the ordinate perpendicular to c A C of the curve A B C represent the condensation in the wave which moves towards the right, in which direction we will suppose x and y measured positively; while the corresponding ordinate of the curve a b c represents the condensation in the wave which moves to the left. Assuming the truth of Boyle's law in cases of motion (which it will suffice to do for the purposes of this paper), if the wave corresponding to ABC (or, as for brevity I shall designate it the wave ABC) stood alone, we should have dp dx positive through out the hinder (left hand) half of the wave, and negative throughout the front half; and as the velocity in A B C is throughout positive, it follows that throughout the hinder half of A B C we dp dy shall have positive, while throughout the front half of ABC dx dt the same function will be negative. For the entire wave, therefore, it is clear, from the symmetrical form of vibration which we have ascribed to it, that the sum of the tension will be zero*. If we now consider the wave a b c taken singly, we shall have throughout its hinder (right hand) half negative, while dx dp throughout its front half de will be positive; and the velocity dx throughout a b c being negative, we shall have dp dy dy positive dx dt throughout the hinder half of the wave, and negative throughout the front half. Thus, in the case of either wave taken separately, the positive and negative parts of the term in the equation of vis viva depending on the tensions will counterbalance each other, and the tensions will wholly disappear from that equation. But when the waves are superposed, or interfere, as, for instance, in the manner represented in the figure, this mutual balance of the opposing terms will cease to exist, as I shall now proceed to show. The figure is supposed to represent the state of things occurring after the period of complete occultation, when the front half of each wave has entirely emerged, and while the hinder halves have in part emerged, but as to the remainder are still in the state of superposition. Draw p P M perpendicular to c A C, and on the left of D, the point of intersection of the curves ABC, a b c. When the waves are separate and non-interferent, a stratum of the hinder half of either wave of the undisturbed breadth da, in the case of the one corresponding to PM, in the other to p m, would give rise to a positive term in the equation of vis viva. But when these elementary portions of the two waves become superposed, although the condensation of the combination will be the sum of the condensations of the two elements taken separately, yet, inasmuch as the values of for the separate elements dp dac for the combination may be dx * Though less obvious, the same is equally true whatever be the form of vibration, provided that the wave is such as to be transmitted without undergoing change in its length, or form of vibration. Phil. Mag. S. 4. Vol. 49. No. 326. May 1875. 2 D dp dx positive or negative according to circumstances. If the sign of in the combination be positive, since it will be multiplied in the equation of vis viva by a negative velocity (for the velocity in the combination will be the difference of the velocities of the components, and the velocity in abc is here predominant), we dp dy shall have at this point dx dt negative; so that this portion of the wave, instead of aiding to balance the negative tensions prevailing throughout the front halves of the two waves, as its components would have done if the two waves had continued separate, will, so to speak, go over to the side of the latter. dp dx On the other hand, if at this point of the combined disturbance dp dx be negative, its value will be less in amount, irrespective of sign, than what it would have been for the wave a b c taken separately, at the same time that, as before, it will be multiplied by a negative velocity, but a velocity which will be less than the velocity with which would be multiplied if we were dealing with the wave abc separately. Hence, though at this point the element of the combined disturbance would, as in the case of each of its components taken separately, tend to counteract the negative tensions of the emerged front halves of the two waves, yet it would do this in a less degree than one only of those components would do when taken separately (the wave a bc to wit), and therefore, à fortiori, in a less degree than both. If we had drawn p P M on the right side of D, we should have arrived at precisely the same conclusion, though in a slightly different manner. It thus appears that, while the negative tensions at the time represented in the figure are precisely the same as when the waves were separate, the positive portion of the tensions will be diminished in amount, so that they will no longer counterbalance the former. On the whole, therefore, the sum of the tensions, instead of being zero, will give rise to a negative term of finite magnitude in the equation of vis viva. III. If it should be supposed that the fact of the sum of the tensions not vanishing in the equation of vis viva may afford a possible source of compensation for the loss of vis viva which it has been shown may arise from the interference of waves, the foregoing investigation will suffice to show the fallacy of this For, in the case of motion above considered it is evident that the destruction of the vis viva may be accompanied by the development of a negative term due to the tensions; in which case there will be a loss of energy, not only through the destruc view. tion of vis viva, but by reason of the fact that what vis viva is left will be more or less counteracted by the presence of the negative term due to the tensions. It is clear, therefore, that the conservation of vis viva cannot be relied on as holding in cases of the intersection of waves traversing continuous masses, and that what Dr. Helmholtz has offered to our attention as a universal law of nature is completely contradicted by fact in cases such as those we have been considering. If we advert to the original derivation of the equation of vis viva from the principle of virtual velocities, if we reflect how completely the arbitrary displacement of the points of application of the respective forces forms the characteristic feature of the principle of virtual velocities, and bear in mind that the transition from that principle to the equation of vis viva is effected simply by the substitution of the actual for the arbitrary motions-if we keep in view these various considerations, it can hardly be matter of surprise that, in treating of the internal metions of continuous masses it should have come to be considered that the only terms depending on the internal actions which need be taken into account in forming the equation of vis viva consist of pairs of equal and opposite forces multiplied by a common displacement or common velocity (that, namely, of the common point of application of the forces), and consequently that no such terms will appear in that equation. That this mode of considering the subject, however specious, must be entirely erroneous, sufficiently appears, if I mistake not, from what has preceded; but it is of the highest importance that the mode as well as the fact of the fallacy should be clearly apprehended; and to this part of the subject I propose now to address myself. P2 z Let P, P2 be the pressures, and v,, v, the particle-velocities at the time t at the surfaces the ordinates of whose positions of rest are respectively x+dx and x+2dx; p-1, P-2, V-1, V-2 the corresponding quantities for the surfaces as to which - dx and x-2dx are the ordinates of the positions of rest. The equation of motion of the first element may be put under the form |